Question

Find the condition that x3 – px2 + qx – r = 0 may have the roots in G.P.

Answer 1

Let , a/R, a, aR be the roots of x3 – px2 + qx – r = 0

Then product of the roots = a/R .a. aR = a3 = r ⇒ a = r1/3

∵ a is a root of x3 – px2 + qx – r = 0

⇒ a3 – pa2 + qa – r = 0

But a = r1/3

⇒ (r1/3)3 – p(r1/3)2 + q(r1/3) – r = 0

⇒ r – p. r2/3 + q. r1/3 – r = 0 ⇒ p. r2/3 = q r1/3

By cubing on both sides

⇒ p3r2 = q3r

⇒ p3r = q3 is the required condition.